A116385 Expansion of e.g.f. Bessel_I(2,2x) + 2*Bessel_I(3,2x) + Bessel_I(4,2x).

0, 0, 1, 2, 5, 10, 21, 42, 84, 168, 330, 660, 1287, 2574, 5005, 10010, 19448, 38896, 75582, 151164, 293930, 587860, 1144066, 2288132, 4457400, 8914800, 17383860, 34767720, 67863915, 135727830, 265182525, 530365050, 1037158320, 2074316640

Offset: 0

Author

Paul Barry, Feb 12 2006

Keywords

• easy
• nonn

Comments

Third column of the Riordan array A116382.

Apart from its root term -1: central terms of the triangle in A051631: a(n) = A051631(n+1, [(n+1)/2]). - Reinhard Zumkeller, Nov 13 2011

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
• Gennady Eremin, Dyck Numbers, II. Triplets and Rooted Trees in OEIS A036991, arXiv:2211.01135 [math.NT], 2022.

Crossrefs

Cf. A001405.

Programs

Haskell

a116385 n = a051631 (n+1) $ (n+1) `div` 2
-- Reinhard Zumkeller, Nov 13 2011

Mathematica

With[{nn=40},CoefficientList[Series[BesselI[2,2x]+2BesselI[3,2x]+ BesselI[ 4,2x],{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Sep 14 2011 *)

PARI

a(n)= binomial(n+3, (n+3)\2) - 3*binomial(n+1, (n+1)\2) \\ Bill McEachen, Dec 12 2022

Formula

E.g.f.: (d/dx)(Bessel_I(3,2x),x) + 2*Bessel_I(3,2x).

a(n) = C(n+1,floor((n-2)/2))*(1+(-1)^n)/2 + C(n,floor((n-3)/2))*(1-(-1)^n).

Conjecture: (n+4)*a(n) -2*a(n-1) +(-7*n-8)*a(n-2) +6*a(n-3) +12*(n-2)*a(n-4)=0. - R. J. Mathar, Jun 13 2014

a(n) = A001405(n+3) - 3*A001405(n+1) (from Eremin link). - Bill McEachen, Dec 12 2022

G.f.: (-1 - x + x^2 + B(x) - 3*x^2*B(x))/x^3, where B(x) is the g.f. of A001405. - Gennady Eremin, Oct 09 2023